Abstract

We propose an approach for showing rationality of an algebraic variety X. We try to cover X by rational curves of certain type and count how many curves pass through a generic point. If the answer is 1, then we can sometimes reduce the question of rationality of X to the question of rationality of a closed subvariety of X. This approach is applied to the case of the so-called Ueno-Campana manifolds. Assuming certain conjectures on curve counting, we show that the previously open cases X_4,6 and X_5,6 are both rational. Our conjectures are evidenced by computer experiments. In an unexpected twist, existence of lattices D₆, E₈, and ${\Lambda }_{10}$ turns out to be crucial.

摘要

我们提出了一种显示代数簇X合理性的方法。我们尝试用某种类型的有理曲线覆盖X，并计算有多少条曲线通过一个通用点。如果答案是 1，那么我们有时可以将 X 的合理性问题简化为X的封闭子变体的合理性问题。这种方法适用于所谓的 Ueno-Campana 流形的情况。假设关于曲线计数的某些猜想，我们表明先前打开的案例X _4,6和X _5,6都是合理的。我们的猜想得到了计算机实验的证实。在一个意想不到的转折中，格子D _6的存在, E ₈ , 和${\Lambda }_{10}$事实证明是至关重要的。